`Chapter 14Moments and torque14.1MSummary of moments and torquesDescriptionnExpressionFQ /O= r Q/O × FQ i=1 MS/OMoment of force FQ about point O Moment of a set S of forces about point O Shift theorem for moment of a force Torque is the moment of a set S of forces whose resultant is 0.MS/O = MS/P = TS =MFQi /O+ r O/P × FSMS/O14.2Moment of a vectorQFQThe moment of a vector results from the cross product of a position vector with a bound vector.a The moment of the vector FQ (bound to point Q) about point O is Q denoted MF /O and is defined in terms of r Q/O (Q's position vector from O) asar Q/OO(1)A bound vector is a vector that is bound to a point.MFQ /O= r Q/O × FQ14.2.1Moment arm of a vector about a point QThe moment arm of the bound vector FQ about point O is the distance d between O and the line of actiona of FQ and measures the effectiveness of FQ at creating a moment about O. This distance can be calculated in various ways, e.g, using the unit vector u in the direction of FQ and/or the angle  between u and r Q/O . d =aFQdr Q/OOrQ/Osin() =rQ/O×uQ=r Q/O · r Q/O-(r Q/O · u)2QMFQ /O= FQ dThe line of action of the bound vector Fis the line passing through Q and parallel to F .14.2.2Moment of a set of vectorsMS/O = MS/O = n i=1The moment of a set S of bound vectors FQ1 , ..., FQn about a point O is defined as the sum of the moments of each bound vector in equation (2). In view of Newton's law of action/reaction in Section 13.5.5, the moment of a system S's internal forces is 0 and MS/O can be written much more simply as solely the moment of external forces on S. 141MFQi /O(2) (3)(13.7)MS/Oexternal14.2.3Statics, dynamics, and the moment of a set of forcesThe moment MS/O of all forces on a system S about an arbitrary point O is related to the time-derivative of S's angular momentum about O in N and other quantities. This relationships simplifies to static equilibrium in equation (4) when the system S is at rest (not moving) in N or S is considered massless.OMS/O(statics)=0(4)14.2.4Shift theorem for the moment of a set of vectorsMS/P , the moment of a set S of bound vectors about a point P , can be calculated in terms of: · MS/O , the moment of S about a point O · r O/P , O's position vector from P · FS , the resultant of SPr O/PO(5)MS/P = MS/O + r O/P × FS14.2.5Torque of a set of vectorsTorque is the moment of a set S of vectors whose resultant is zero. TS = MS/OwhereFS = 0and point O is any point(6)Since a couple is a set of vectors whose resultant (sum) is 0, a torque is the moment of a couple.1 A couple has a special property, namely, the moment of a couple about a point O is equal to the moment of the couple about any other point Q. As a result, a torque is not associated with a point. The following example highlights the difference between a torque and a moment. Consider the various sets S of forces in Figure ??, and fill in the following table by calculating FS , the resultant of S, and MS/O , MS/P , MS/Q , the moments of S about points O, P , and Q, respectively. Express your results in terms of the right-handed orthogonal unit vectors nx , ny , nz . S A B C D FS 10 ny MS/O 50 nz MS/P 0 MS/Q MS/O = MS/P = MS/Q ? Yes/No Yes/No Yes/No Yes/No Moment is a torque? Yes/No Yes/No Yes/No Yes/No14.2.6Torque on a rigid body (or reference frame)Torques can be associated with reference frames when the vectors that comprise the torque have a special character. The notation TA is used to designate a torque of a couple whose vectors F1 , ..., Fn have lines of actions that pass through points Q1 , ..., Qn , respectively, each of which is fixed in a reference frame A. When Q1 , ..., Qn are also fixed in a second reference frame, e.g., B, then one designates the torque TA/B .Torques are a special type of Moment, namely the moment of a couple. All Torques are Moments, but not all Moments are Torques. A useful analogy is all Toyotas are Motor-vehicles, but not all Motor-vehicles are Toyotas.Copyright c 1992-2009 by Paul Mitiguy1142Chapter 14:Moments and torque10 n10 nAO5mBP3mQO5mP3mQny10 n10 nCO6n 5mnzQ6nnx10 nDO5m 4nP3mP3m 6nQFigure 14.1: Various sets of forces14.3Proof of shift theorem for the moment of a set of bound vectorsTo establish the validity of equation (5), consider a set S of bound vectors F1 , ..., Fn , whose lines of action pass through points Qi (i = 1, ..., n), respectively. The moment of S about a point P is defined in terms of r Qi /P (Qi 's position vector from P ) as MS/P = n i=1(2)r Qi /P × FQi(7)Qi 's position vector from P may be written as r Qi /P = r O/P + r Qi /O Substituting equation (8) into equation (7), distributing the summation, and rearrangement yields MS/Pn (7,8)(8)= =r O/P × FQi +n i=1n i=1 n i=1r Qi /O × FQi r Qi /O × FQi (9)i=1r O/P ×FQ i +The first summation in equation (9) is the definition of FS (the resultant of S). The second summation in equation (9) is the definition of MS/O (the moment of S about O). Combining these two facts produces equation (5), the shift theorem for the moment of a set of bound vectors, namely MS/P = r O/P × FS + MS/OCopyright c 1992-2009 by Paul Mitiguy143Chapter 14:Moments and torque`

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